Or: what is the "greenhouse effect" physically?

The Case of the Missing 33 Kelvin Continued

Last time, when we talked about putting the Earth in a box, we saw that a simple back-of-the-envelope calculation of the energy balance and the resulting black body temperature of the earth comes surprisingly close to the right answer. But there is a gap: the black body temperature calculated with a zero dimensional energy balance model is about 33 kelvin lower than the estimated average surface temperature on earth.

In other words, our model predicts an Earth that’s 33 °C colder than it really is!

In such a situation, as theoretical physicists, we start by congratulating ourselves on a successful first approximation, and then look out for the next most important effect that we need to include in our model.

This effect needs to:

1) have a steady and continuous influence over thousands of years,

2) have a global impact,

3) be rather strong, because heating the planet Earth by 33 kelvin on the average needs a lot of power.

The simplest explanation would of course be that there is something fundamentally wrong with our back-of-the-envelope calculation.

One possibility, as Itai Bar-Natan mentioned last time, is geothermal energy. It certainly matches point 1, maybe matches point 2, but it is hard to guess if it matches point 3. As John pointed out, we can check the Earth’s energy budget on Wikipedia. This suggests that the geothermal heating is very small. Should we trust Wikipedia? I don’t know. We should check it out!

But I will not do that today. Instead I would like to talk about the most prominent explanation:

Most of you will of course have heard about the effect that climate scientists talk about, which is often - but confusingly - called the ‘greenhouse effect’, or ‘back radiation’. However, the term that is most accurate is downward longwave radiation (DLR), so I would like to use that instead.

In order to assess if this is a viable explanation of the missing 33 kelvin, we will first have to understand the effect better. So this is what I will talk about today.

In order to get a better understanding, we will have to peek into our simple model’s box and figure out what is going on in there in more detail.

Peeking into the Box and Splitting up: Surface and Atmosphere

To get a better approximation, instead of treating the whole earth as a black body, we will have to split up the system into the Earth itself, and its atmosphere. For the surface of the Earth it is still a good approximation to say that it is a black body.

The atmosphere is more complicated. In a next approximation step, I would like to pretend that the atmosphere is a body of its own, hovering above the surface of the earth, as a separate system. So we will ignore that there are several different layers in the atmosphere doing different things, including interactions with the surface. Well, we are not going to ignore the interaction with the surface completely, as you will see.

Since one can quickly get lost in details when discussing the atmosphere, I’m going to cheat and look up the overall average effects in an introductory meteorology textbook:

Here is what atmosphere and Earth’s surface do to the incoming radiation from the Sun (from page 48):

Of 100 units of inbound solar energy flux, 30 are reflected or scattered back to space without a contribution to the energy balance of the Earth. This corresponds to an overall average albedo of 0.3 for the Earth.

The next graphic shows the most important processes of heat and mass transport caused by the remaining 70 units of energy flux, with their overall average effect (from page 49):

Maybe you have some questions about this graphic; I certainly do.

Conduction and Convection?

Introductory classes for partial differential equations sometimes start with the one dimensional heat equation. This equation describes the temperature distribution of a rod of metal that is heated on one end and kept cool on the other. The kind of heat transfer occurring here is called conduction. The atoms or molecules stay where they are and transfer energy by interacting with their neighbors.

However, heat transfer by conduction is negligible for gases like the atmosphere. Why is it there in the graphic? The answer may be that conduction is still important for boundary layers. Or maybe the author wanted to include it to avoid the question “why is conduction not in the graphic?” I don’t know. But I’ll trust that the number associated with the “convection and conduction” part is correct, for now.

What is Latent Heat?

There is a label “latent heat” on the left part of the atmosphere: latent heat is energy input that does not result in a temperature increase, or energy output that does not result in a temperature decrease. This can happen, when there is a phase change of a component of the system. For example, when liquid water at 0°C freezes, it turns into ice at 0°C while losing energy to its environment. But the temperature of the whole system stays at 0°C.

The human body uses this effect, too, when it cools itself by sweating. This cooling effect works as long as the fluid water turns into water vapor and withdraws energy from the skin in the process.

The picture above shows a forest with water vapor (invisible), fluid (dispersed in the air) and snow. As the Sun sets, parts of the water vapor will eventually condense, and fluid water will turn into ice, releasing energy to the environment. During the phase changes there will be energy loss without a temperature decrease of the water.

Downward Longwave Radiation: The Atmosphere is not a Black Body

When there is a lot of light there are also dark shadows. — main character in Johann Wolfgang von Goethe's Götz von Berlichingen

Last time we pretended that the Earth as a whole behaves like a black body.

Now that we split up the Earth into surface and atmosphere, you may notice that:

a) a lot of sunlight passes through the atmosphere and reaches the surface, and

b) there is a lot of energy flowing downwards from the atmosphere to the surface in form of infrared radiation. This is called downward longwave radiation.

Observation a) shows that the atmosphere does not act like a black body at all. Instead, it has a nonzero transmittance, which means that not all incoming radiation is absorbed.

Observation b) shows that assuming that the black body temperature of the Earth is equal to the average surface temperature could go wrong, because—from the viewpoint of the surface—there is an additional inbound energy flux from the atmosphere.

The reason for both observations is that the atmosphere consists of various gases, like O2, N2, H2O (water vapor) and CO2. Any gas molecule can absorb and emit radiation only at certain frequencies, which are called its emission spectrum. This fact led to the development of quantum mechanics, which can be used to calculate the emission spectrum of any molecule.

Molecules and Degrees of Freedom

When a photon hits a molecule, the molecule can absorb the photon and gain energy in three main ways:

• One of its electron can climb to a higher energy level.

• The molecule can vibrate more strongly.

• The molecule can rotate more rapidly.

To get a first impression of the energy levels involved in these three processes, let’s have a look at this graphic:

The y-axis shows the energy difference in ‘eV’, or ‘electron volts’. An electron volt is the amount of energy an electron gains or loses as its potential changes by one volt.

Accoding to quantum mechanics, a molecule can emit and absorb only photons whose energy matches the difference of one of the discrete energy levels in the graphic, for any one of the three processes.

It is possible to use the characteristic absorption and emission properties of molecules of different chemical species to analyze the chemical composition of an unknown probe of gases (and other materials, too). These methods are usually called names involving the word ‘spectroscopy’. For example, infrared spectroscopy involves methods that examine what happens to infrared radiation when you send it to your probe.

By the way, Wikipedia has a funny animated picture of the different vibrational modes of a molecule on the page about infrared spectroscopy.

But why does so much of radiation from the Sun pass through the atmosphere, while a lot of infrared radiation emitted by the Earth instead bounces back to the surface? The answer to this puzzle involves a specific property of certain components of the atmosphere.

Can you see an electron hopping?

Here is a nice overview of the spectrum of electromagnetic radiation:

The energy EE and the wavelength λ\lambda of a photon have a very simple relationship:

Plugging these into the formula we get that a photon with an energy of one electron volt has a wavelength of about 1.21.2 micrometers, which is just outside the visible range, a bit towards the infrared direction. The visible range corresponds to 1.6 to 3.4 electron volts. If you want, you can scroll up to the graphic with the energy levels and calculate which processese will result in which kind of radiation.

Electrons that take a step down the orbital ladder in an atom emit a photon. Depending on the atom and the kind of transition, some of those photons will be in the visible range, and some will be in the ultraviolet.

There is no Infrared from the Sun (?)

From the Planck distribution, we can determine that the Sun and Earth, which are approximately black bodies, emit radiation mostly at very different wavelengths:

This graphic is sometimes called ‘twin peak graph’.

Oversimplifying, we could say: The Earth emits infrared radiation; the Sun emits almost no infrared. So, if you find infrared radiation on earth, you can be sure that it did not come from the Sun.

The problem with this statement is that, strictly speaking, the Sun does emit radiation at wavelengths that are in the infrared range. This is the reason why people have come up with the term near-infra-red radiation, which we define to be the range of 0.85 and 5.0 micrometer wavelength. Radiation with longer wavelengths is called far infrared. With these definitions we can say that the Sun radiates in the near-infra-red range, and earth does not.

Only certain components of the atmosphere emit and absorb radiation in the infrared part. These are called—somewhat misleadingly—greenhouse gases. I would like to call them ‘infrared-active gases’ instead, but unfortunately the ‘greenhouse gas’ misnomer is very popular. Two prominent ones are H2O and CO2:

The atmospheric window at 8 to 12μm is quite transparent, which means that this radiation passes from the surface through the atmosphere into space without much ado. Therefore, this window is used by satellites to estimate the surface temperature.

It is not a coincidence that molecules made of different kinds of atoms like CO2 react to infrared radiation, while those with one kind like O2 do not. The deeper reason is that molecules with different kinds may have a so-called ‘dipole moment’.

Since most radiation coming from the Earth is infrared, and only some constituents of the atmosphere react to it—excluding the major ones—a small amount of, say, CO2could have a lot of influence on the energy balance. Like being the only one in a group of hundreds with a boom box. But we should check that more thoroughly.

Can a Cold Body Warm a Warmer Body?

Downward longwave radiation warms the surface, but the atmosphere is colder than the surface, so how can radiation from the colder atmosphere result in a higher surface temperature? Doesn’t that violate the second law of thermodynamics?

The answer is: no, it does not. It turns out that others have already taken pains to explain this on the blogosphere, so I’d like to point you there instead of trying to do a better job here:

It's the Numbers, Stupid!

Shut up and calculate! — Leitmotiv of several prominent physicists after becoming exhausted by philosophical discussions about the interpretation of quantum mechanics.

Maybe we have succeeded by now to convince the imaginary advisory board of the zero dimensional energy balance model project that there really is an effect like ‘downward longwave radiation’. It certainly should be there if quantum mechanics is right. But I have not explained yet how big it is. According to the book Meteorology Today, it is big. But maybe the people who contributed to the graphic got fooled somehow; and there really is a different explanation for the case of the missing 33 kelvin.

What do you think?

When we dip our toes into a new topic, it is important to keep simple yet fundamental questions like this in mind, and keep asking them.

In this case we are lucky: it is possible to measure the amount of downward longwave radiation. There are a lot of field studies, and the results have been incorporated in global climate models. But we will have to defer this story to another day.